6 problems

Consider the proposed quadrature method applied to the nonlocal problem on a non-uniform discretization, and measure the numerical error in the $L^2$ norm. Convergence-rate conject…

Consider the proposed quadrature approach for two-dimensional non-uniform discretizations, constructed as tensor products of perturbed one-dimensional meshes. Let the convergence r…

Localized cancellation conjecture. The approximate ${\mathcal O}(h^2)$ convergence observed on such grids holds because the beneficial cancellation effect is a localized phenomenon…

Seven-point quadrature conjecture. The seven-point rule consists of sufficiently many quadrature points to produce stable second-order convergence rates for the exact ball and for…

Geometric cancellation conjecture. The cancellation between areas added to and subtracted from the exact ball may yield

Sharpness conjecture. The constant $\frac{\pi}{2\sin(\pi s)}$ in this inequality is sharp, at least for $n\geq 3$. The estimate is established in the stated parameter range before…